*3.1. Experimental Setup and Polymer Sheets*

#### 3.1.1. Setup

The SPIF process is performed using an ALECOP-ODISEA conventional milling machine, with an in-house developed fixing system (Figure 4a). The sheet fixing system is placed on the machining bed, and it consists of a frame made of four aluminum profiles, a die with a hole of 140 mm diameter and an upper die. Eight screws are used to fasten the two dies. Two different tools made of aluminum, with a hemispherical tip of 10 and 12 mm in diameter are used. A lubricant fluid has been used during the experiments.

A 9257BA Kistler dynamometer table is used to measure the value of the force. The dynamometer table is placed between the milling table and the fixing system.

A Flir T335 thermal imaging camera, with a 320 × 240 pixel resolution, is used to measure the temperature reached in the polymer sheet during the process.

Two 200 × 200 mm PVC and PC sheets with a thickness of 3 mm are used. The maximum size of the sheet is the same than that of the working space of the CNC machine to properly fix the sheet. The final shape of the formed sheet is expected to have a 40 mm deep cone, an outer diameter of 128 mm and a cone opening angle *α* with values between 45◦ and 60◦. Figure 4b shows the shape of the specimen and its theoretical dimensions.

**Figure 4.** Setup for SPIF: (**a**) CNC machine with fixing system; (**b**) Shape of processed sheet.

#### 3.1.2. Material Properties

The monotonic stress-strain behavior of both glassy polymers, PVC and PC, at different temperatures were characterized by universal testing machine with 6 kN load cell to calibrate the constitutive model, taking into account the temperature material dependence during the simulation. The tests were carried out in uniaxial tension with three replications of each test by following the ASTM D 638-02a norm. The tests were performed at constant strain rate of 1000 mm/min at 273 and 373 K for PC specimens and 273, 313 and 343 K for PVC specimens. It was noticed that both materials experienced a linear elastic response followed by a yielding, after this point the material undergoes softening behavior (Figure 5). The parameters used for the above described material model (Table 2) were calibrated by Mcalibration® commercial software. Instead of an exponential law, a stress-strain tabular data is used to define the evolution of the yield surface size σ0. In order to fit PVC and PC material parameters at different temperatures, an optimization method based on Nelder-Mead algorithm was employed. The fitness function was the coefficient of determination *R*2, Figure 5 shows the coefficient *R*<sup>2</sup> for all curves. The calibration was performed with rate independence, for this reason only yield stress values and equivalent plastic strain are provided.

In this work, the experimental tests were conducted with lubricant fluid and, therefore, a low temperature variation was noted. Although it is not the aim of this work, once the numerical simulation is validated, it could be used to predict the temperature evolution since the material model was fitted for different temperatures. Figure 6 shows an example of temperature prediction, the gradient of temperature localized in the plastic sheet along the wake generated by the steel tip was due to the dissipated energy converted into heat due to the frictional sliding. This energy is responsible of the increase of temperature in the thermoplastic sheet that requires a temperature-dependent material model to predict the axial force relaxation.

**Figure 5.** Experimental behavior and calibrated model results at different temperatures for PC (**a**) and PVC (**b**).

**Table 2.** Parameters of the material model proposed.


**Figure 6.** Temperature for a PVC sheet: (**a**) Example of experimental temperature measurements; (**b**) numerical temperature estimation.

### *3.2. Experimental Procedures*

One SPIF test for tuning the numerical model was performed. On the other hand, three tests, changing the deformed volume by means of process parameters, were carried out to determine the semi-analytical approximation. These tests are repeated for each material considered (PVC and PC). In order to validate the models, two additional tests with different conditions to that of the aforementioned tests were carried out.

#### **4. Results and Discussion**

#### *4.1. Tuning Tests*

The measured axial forming force profile and the numerical estimation, for the tuning tests described in Table 3, are shown in Figure 7. There is an initial transition zone where the forming force grows and a stationary region (*z* > 15 mm) where the force stands approximately constant. The numerical model was fitted to reproduce the stationary region, and the final numerical result is portrayed in Figure 7. For each *z*, the numerical force is the mean of three values obtained at random punch positions, which can be the reason for the oscillation observed in the estimated curve.

**Table 3.** Experimental tests conducted to define the semi-analytical and numerical model.

**Figure 7.** Numerical and experimental axial force profile. (**a**) PC process conditions: Cone shape, *α* = 60◦, *D* = 12 mm and Δ*z* = 0.8 mm; (**b**) PVC process conditions: Cone shape, *α* = 60◦, *D* = 10 mm and Δ*z* = 0.6 mm.

#### *4.2. Specific Energy Equations*

Figure 8 shows the forces obtained by the semi-analytical model tests described in Table 3. These forces are the mean value of the axial forming force in the stationary region of each of the experimental tests. Except for the intercept, the experimental measurements agree with Equation (2). The intercept can be interpreted as a minimum forming force to obtain a local deformation.

By fitting the experimental data using a linear function, an equation can be obtained that predicts the value of the axial force as a function of the deformed volume and the step down:

$$Fz = \mathcal{U} \cdot \frac{V}{\Delta z} + F\_0 \tag{13}$$

where *U* is the specific forming energy and *F*<sup>0</sup> is the minimum forming force. For each of the tested polymers, different values of the specific forming force and the minimum force were obtained.

The high value of the determination coefficient shows a good agreement between experimental measurements and the linear approximation.

**Figure 8.** Axial forming forces approximations for PC and PVC.

Only three experimental measurements are enough to obtain the characteristic curve for a specific material. With this curve, any other contact conditions for a material can be computed, predicting the axial forming forces in a simple and accurate way.

#### *4.3. Additional Validation Tests*

In order to validate the numerical and semi-analytical models, a test using a different condition to the previous tests was carried out. Figure 9 shows the relative error of the estimation models respect to the stationary axial forming force measurements. The two estimation models agree with the actual forces, and have a relative error below 10%. The best result obtained with the semi-analytical model is due to the deformed volume (in this test) is within the fitted region.

The numerical model also provides wall thickness reduction respect to the radial distance (Figure 10). Numerical and experimental thickness measurements are similar for both PVC and PC sheets. The percentage of the mean relative error is always lower than 6% in both cases.

**Figure 9.** Relative error of the estimation for additional tests. Process conditions: Cone shape, *α* = 60◦, *D* = 10 mm and Δ*z* = 0.8 mm.

**Figure 10.** Thickness evolution respect to radial distance, (**a**) PVC and (**b**) PC. Process conditions: Cone shape, *α* = 60◦, *D* = 10 mm and Δ*z* = 0.8 mm.

Minimum thickness values are correctly predicted for both polymers. For the PVC sheet, the numerical model also estimates the radial distance where the minimum thickness occurs (maximum stretching). SPIF process uniformly pushes the same percentage of material to the strain direction. At both wall ends, the thickness is greater than that at the initial component wall: Initial thickness is reduced through the wall by stretching the material to the bottom of the component.

#### **5. Conclusions**

The study of forming force in SPIF is key to improve the process. Poor dimensional accuracy is an important SPIF drawback that can be treated knowing the forming force. There are several analytical, experimental and numerical approaches to determine the forming force but, in general, the existing solutions are focused on metals. For polymers, which also have interesting potential applications, it is difficult to accomplish forming force predictions because of their more complex material behavior.

This work shows two axial forming forces approaches: (1) A simple method based on experimental measurements for different deformation volumes and materials, which extends the specific energy concept used in cutting models to incremental forming processes; and (2) a numerical method, that implements a hyperelastic-plastic material.

Material property curves and the axial forming force at steady state for a set of process conditions were used to tuning the numerical model. Regarding to the semi-analytical solution, we conducted experiments with different axial depth, tool diameter and forming angle in order to study the deformation volume influence on the forming force. These tests were repeated for two polymers PVC and PC.

Once the numerical and the semi-analytical models were ready, we compared their estimations with the real measurements of additional forming tests. The results agree with the experimental measurements, a percentage of relative error below 10% was obtained.

Extending the semi-analytical procedure to more materials, and further works on this topic could lead to a fast and robust forming force estimation model. On the other hand, the numerical model not only obtained a good forming force prediction, but also additional variables such as thickness evolution estimation, which together with forming force are two of the main variables in any mechanical forming process.

The two approaches could help to improve the knowledge of the SPIF process on polymers. Each solution addresses a research need: Fast computation (semi-analytical model), or low experimental cost (numerical model).

**Author Contributions:** Conceptualization, G.M.-S., A.G.-C. and R.D.-V.; Methodology, G.M.-S., A.G.-C. and R.D.-V.; Software, G.M.-S., A.G.-C. and R.D.-V.; Validation, G.M.-S., A.G.-C. and R.D.-V.; Formal Analysis, G.M.-S., A.G.-C. and R.D.-V.; Investigation, G.M.-S., A.G.-C. and R.D.-V.; Resources, G.M.-S., A.G.-C. and R.D.-V.; Data Curation, G.M.-S., A.G.-C. and R.D.-V.; Writing-Original Draft Preparation, G.M.-S., A.G.-C., D.C. and R.D.-V.; Writing-Review & Editing, G.M.-S., A.G.-C., D.C. and R.D.-V.; Visualization, G.M.-S., A.G.-C., and R.D.-V.; Supervision, G.M.-S., A.G.-C., D.C. and R.D.-V.; Project Administration, G.M.-S., A.G.-C., D.C. and R.D.-V.

**Acknowledgments:** The authors would like to thank the "Mechanical and Energy Engineering" TEP 250 research group.

**Conflicts of Interest:** No competing financial interests exist.
